Exact Convex Reformulations of Linear Neural Networks via Completely Positive Lifting

TL;DR

This paper presents an exact convex reformulation of deep linear neural network training problems using completely positive lifting, linking neural network training to copositive programming.

Contribution

It introduces a novel convex reformulation that captures the nonconvexity of linear neural networks through completely positive cones, independent of network depth.

Findings

Reformulation has the same optimal value as the original problem.

The lifted dimension depends only on input/output dimensions, not network depth.

Connects neural network training with copositive programming.

Abstract

We show that the training problem of a deep linear neural network under the squared loss admits an exact convex reformulation in a lifted space over a generalized completely positive cone. The reformulation has the same optimal value as the original nonconvex problem and is linear in the lifted variables, with all nonconvexity encoded in the cone constraint. Its ambient lifted dimension depends only on the input and output dimensions, independent of the network depth and the number of data points, and the bottleneck width enters only through scalar constraints. The construction proceeds by reducing the multilayer parameterization to a bilinear factorization, lifting it to a rank-constrained semidefinite program, expressing the rank constraint via a complementarity condition, and applying a completely positive lifting. While the resulting formulation is computationally intractable in general, it gives an exact conic representation of the nonconvexity induced by linear factorization and connects linear neural network training with copositive programming.